Computing and Optimizing Over All Fixed-Points of Discrete Systems on Large Networks

Abstract

Equilibria, or fixed points, play an important role in dynamical systems across various domains, yet finding them can be computationally challenging. Here, we show how to efficiently compute all equilibrium points of discrete-valued, discrete-time systems on sparse networks. Using graph partitioning, we recursively decompose the original problem into a set of smaller, simpler problems that are easy to compute, and whose solutions combine to yield the full equilibrium set. This makes it possible to find the fixed points of systems on arbitrarily large networks meeting certain criteria. This approach can also be used without computing the full equilibrium set, which may grow very large in some cases. For example, one can use this method to check the existence and total number of equilibria, or to find equilibria that are optimal with respect to a given cost function. We demonstrate the potential capabilities of this approach with examples in two scientific domains: computing the number of fixed points in brain networks and finding the minimal energy conformations of lattice-based protein folding models.

Nonlinear consensus on networks: equilibria, effective resistance and trees of motifs

“…While some networks will consist of a single block, for many other networks this procedure will drastically reduce the dimensions in which we are working. The approach of finding equilibria by studying sub-networks separately is very powerful, as was shown in a different context and for other types of dynamics in [40]. 9 That is, we know their equilibrium points and their stability.…”